Theorems · Definition · category theory

CategoryTheory.exponentialIdealReflective

{C : Type u₁} →
  {D : Type u₂} →
    [inst : CategoryTheory.Category.{v₁, u₁} C] →
      [inst_1 : CategoryTheory.Category.{v₁, u₂} D] →
        (i : CategoryTheory.Functor D C) →
          [inst_2 : CategoryTheory.CartesianMonoidalCategory C] →
            [inst_3 : CategoryTheory.MonoidalClosed C] →
              (A : C) →
                [inst_4 : CategoryTheory.Reflective i] →
                  [CategoryTheory.ExponentialIdeal i] →
                    i.comp ((CategoryTheory.ihom A).comp ((CategoryTheory.reflector i).comp i)) ≅
                      i.comp (CategoryTheory.ihom A)

If D is a reflective subcategory, the property of being an exponential ideal is equivalent to the presence of a natural isomorphism i ⋙ exp A ⋙ leftAdjoint i ⋙ i ≅ i ⋙ exp A, that is: (A ⟹ iB) ≅ i L (A ⟹ iB), naturally in B. The converse is given in ExponentialIdeal.mk_of_iso.

Defined in

Mathlib.CategoryTheory.Monoidal.Closed.Ideal

Cited by

0 results in Mathlib

Foundations

Depth 33 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CartesianMonoidalCategoryCategoryTheory.MonoidalClosedCategoryTheory.ReflectiveCategoryTheory.ExponentialIdeal